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4h

comment 
Prescribed values for the uniform density
Btw, I've just realized a mistake in one of my comments: Strauch & Tóth's paper doesn't look for the minimal lower asymptotic density of a set $X\subseteq{\bf N}^+$ s.t. the ratio set of $X$ is dense in $[0,\infty[$; the answer to this question is $0$, by taking $X$ to be the set of all positive rational primes and having a look at the bottom of p. 155 in W. Sierpiński, Elementary Theory of Numbers, PWN, Warszawa, 1964. What Strauch & Tóth prove is, instead, that $\frac{1}{2}$ is the minimal $\gamma\ge 0$ s.t. the ratio set of $X$ is dense in $[0,\infty[$ whenever ${\sf d}_\ast(X)\ge\gamma$. 
4h

comment 
Prescribed values for the uniform density
However, this doesn't mean much to me, when compared with the length of the proof of the main theorem (viz., Th. 3 on p. 77) in Luca & Porubský's 2005 paper, since the latter is, to my eyes, much stronger (and more constructive) than Mišík's Th. 2. Again, I've not read Mišík's paper carefully (I should, but had no time). Yet, I mentioned your comment to other guys who did it (I believe!), and they replied that they aren't aware of any issue with Mišík's paper. So would you mind to be more specific on what you think is missing in Mišík's proof? I'm very interested, and other peps may be too. 
4h

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Prescribed values for the uniform density
@KevinO'Bryant: Mišík's paper has two main results, Th. 1 (p. 291) & Th. 2 (p. 293): Th. 1 is about the independence of the lower and upper $f$densities, ${\underline{d}}_f$ and ${\overline{d}}^f$ in Mišík's notation, associated with a function $f:{\bf N}^+\to{\bf R}^+$ s.t. $\sum_{n\ge 1}f(n)=\infty$ and $f(n)/\sum_{i=1}^nf(i)\to 0$ as $n\to\infty$, while Th. 2 is about the independence of the upper and lower asymptotic and logarithmic densities. In fact, the proof of Th. 2 is just 2+1/2 pages, since we don't really need Lemma 1 here (even if this is not mentioned in the paper...). [tbc] 
May 20 
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Prescribed values for the uniform density
@KevinO'Bryant: I join Charles in his request. What are you alluding to? I don't claim to have read Mišík's paper carefully, but as far as I can say, there is no problem in there (incidentally, the paper is cited, e.g., in: F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 6780, where no reference is made to any gap or other issue in Mišík's work). 
May 20 
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Prescribed values for the uniform density
@Charles: AFAIK, the result that you credit to Strauch & Toth was 1st proved by Georges Grekos in his thesis (Paris 6, 22 June 1976), and later published in: G. Grekos, Répartition Des Densites Des SousSuites D'Une Suite D'Entiers, J. Number Theory 10 (1978), No. 2, 177191 (in French). Strauch & Tóth's paper is definitely focused on a different problem (what's the minimal lower asymptotic density of a set $X\subseteq\bf N^+$ for which the ratio set of $X$, viz. $\{x/y: x,y\in X\}$, is dense in $[0,\infty[$?), and recovers Grekos' result as a corollary. 
May 20 
comment 
On the independence of lower and upper asymptotic and Banach densities
Here is an old thread where a more difficult question is addressed: mathoverflow.net/questions/103111/…. The thread has also the sketch of an answer by @Anthony Quas (mathoverflow.net/a/103127/16537), but there is no reference to the question stated in the OP there. 
May 20 
awarded  Excavator 
May 20 
revised 
Prescribed values for the uniform density
Fixed a typo in an inequality 
May 20 
suggested  approved edit on Prescribed values for the uniform density 
May 18 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
Fixed an important detail 
May 18 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
Added some further considerations 
May 18 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
Extended the answer to make it fit the topic 
May 18 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
Extended the answer to make it fit the topic 
May 17 
accepted  Orderability of $\langle a,b \mid a^{1} b^m a = b^n\rangle$ and $\langle a,b \mid a^{1} b a^m = b^n\rangle$ 
May 17 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
Removed extras from a previous version of the post 
May 17 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
There was a slight excess of "not" 
May 17 
revised 
Conditions for an analogue of CauchyDavenport for simple groups
added 10 characters in body 
May 17 
answered  Conditions for an analogue of CauchyDavenport for simple groups 
May 17 
comment 
On the independence of lower and upper asymptotic and Banach densities
@Gabriel. I've at least two issues with your answer. First, as you too have remarked, it doesn't answer my question. Second, I'm not so convinced that, after filling in the technical details that you're alluding to, the construction you are suggesting will still be so fairly easy: I might mention a number of situations where constructions involving densities are relatively easy (or even trivial to some degree) as long as the "relevant parameters" are rational, but get significantly more complicated otherwise. In any case, thanks for your contribution to the discussion. 
May 16 
revised 
On the independence of lower and upper asymptotic and Banach densities
A pair was missing 